Perturbations of minimizing movements and curves of maximal slope

Antonio Tribuzio

doi:10.3934/nhm.2018019

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2018, Volume 13, Issue 3: 423-448. Doi: 10.3934/nhm.2018019

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Perturbations of minimizing movements and curves of maximal slope

Antonio Tribuzio

Via della Ricerca Scientifica 1, Rome, 00133, Italy

Received: September 30, 2017

Revised: March 31, 2018

Published: August 2018

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Abstract

We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

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Mathematics Subject Classification: 47J35, 47J30, 35B27, 35K90, 49J05.

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Figure 1. Graphs of the discrete solutions with different values of $\tau$. The smaller jumps of $u^\tau$ correspond to the larger parameter $\beta$

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Figure 2. Pinned motion produced by perturbations diverging in the interval $(1, 2)$

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Figure 3. The graphs represent two discrete solutions for the same value of $\tau$, corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear

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Figure 4. On the left the time chart of $u^\tau$, on the right the plot of $\phi(u^\tau)$, corresponding to perturbations with $\delta = 1$. Note how the motion exits from the lower energy state when $t = 1$

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Figure 5. Graphs as in Figure 4 with $\delta = 8$. It is grater than the critical value $e^2$, indeed when $t = 1$ the motion does not exit the first well

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Figure 6. Graphs for $\delta = 8$. As in Figure 5, the motion does not exit the well when $t = 1$, but when $t = 2$ it does

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Figure 7. Graph for $\delta = 1$. The motion always passes to the very next potential well

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Figure 8. Graphs of a discrete solution passing through two potential wells at every jump discontinuity

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